Question

tan x/2=tanh u/2 , prove that u = log tan (π/ 2 + x/2)

Answer 1

Question :

If tan(x/2) = tanh(u/2), prove that

u = log tan(π/4 + u/2).

Proof :

Given, tan(x/2) = tanh(u/2)

⇒ tan(x/2) = $\frac{sinh(u/2)}{cosh(u/2)}$

⇒ tan(x/2) = $\frac{e^{u/2}-e^{-u/2}}{e^{u/2}+e^{-u/2}}$

⇒ tan(x/2) = (eᵘ - 1)/(eᵘ + 1)

⇒ eᵘ - 1 = (eᵘ + 1) tan(x/2)

⇒ eᵘ - 1 = eᵘ tan(x/2) + tan(x/2)

⇒ {1 - tan(x/2)} eᵘ = 1 + tan(x/2)

⇒ eᵘ = {1 + tan(x/2)}/{1 - tan(x/2)}

⇒ eᵘ = $\frac{tan\frac{\pi}{4}+tan\frac{x}{2}}{1-tan\frac{\pi}{4}tan\frac{x}{2}}$

⇒ eᵘ = tan(π/4 + x/2)

⇒ u = log tan(π/4 + x/2)

Hence, proved.

Rules :

• tanh(x) = {sinh(x)}/{cosh(x)}

• sinh(x) = (eˣ - e⁻ˣ)/2

• cosh(x) = (eˣ + e⁻ˣ)/2

• tan(A + B) = (tanA + tanB)/(1 - tanA tanB)

• eˣ = k ⇒ x = logₑk